56 A Discourse on the Theory of Fluxions. 



incremeht of a:" to the increment of a: will he nx^'^z+n. 



n 1 ji — \ Ji 2 



— --x"-*z*+w.— —x'^'^z^-^-^LQ. : z. Dividing by z 



n—l_ n—\ n—2 

 it becomes wa;" *+w.— ^x" '^z+n.— ^^'^" ^^^^c.:!. 



If 2: be diminished by an indefinite subdivision, the increment 

 will approach continually towards nx'^~ ' as its limit. Sup- 

 pose z to be less than any assignable quantity, it will then 

 become equal to o, and all the terms, in which it is found, 

 will vanish. Hence the ultimate ratio of the increments is 

 jix"~^ : 1, or nx''~^x' ; x'. If n=4, then x"=x'^, the ratio 

 of the increments is 4:X'^-\-Gx^z-\-xz'^-\-z^ I 1, and the ulti- 

 mate ratio is 4^^ : 1, or 4x^x' ; x'. Fluxions illustrate the 

 theory by extension and motion, properties which are singu- 

 larly adapted to explain the nature of those quantities, to 

 which fluxions are applied, and which, when divested of the 

 consideration of matter, are with propriety introduced into 

 pure mathematics. Thus the two illustrious inventors of 

 this science have each taken tenable ground in their mode 

 of explaining it, and have placed this branch of the mathe- 

 matics on a foundation which cannot be shaken, and which 

 time will never demolish. Each method has its pecuhar ex- 

 cellencies ; and if either were wanting, the theory would in 

 some respects, be deficient. 



But an illustration of the manner in which fluents are gen- 

 erated, and an explanation of the nature of that relation, 

 which fluxions bear to their fluents, are two distinct things, 

 which ought not to be blended together. Whilst the former 

 is accomplished in a satisfactory manner, the latter remains, 

 in my opinion, unexplained. But some mathematicians have 

 thought differently, and have supposed that the properties of 

 the ratio na;"~^.r* : x' are sufficient to develop the nature 

 of this relation. That this is their view, is manifest from their 

 supposing, that the foundation of fluxions is laid in the tacit 

 acknowledgment, that a circle is fi polygon of an infinite 

 number of sides, (Brewster's Encyclopoedia, Art. Fluxions.) 

 It was upon this supposition, that Carnot admitted that an 

 error actually arises from the rejection of the quantity, which 

 is the difference between the increment and the fluxion ; but 

 that this error in the course of the operation, is compensated 

 by an error of a contrary nature, (Tilloch's Phil. Mag. Vol. 

 8 and 9.) By thus applying principles, which are insufficient 



